Electrical – How to correctly calculate the Rfb value in AD5933

Euler is your friend. Euler's Formula:

\$ e^{j \theta} = cos(\theta) + j \cdot sin(\theta) \$

Euler helps you to calculate in an easy way with the complex impedance, by using the \$e\$ power. At the end of the calculation you can separate the complex power of \$e\$ into its real and imaginary parts. These agree with your resistance and reactive impedance resp.

\$ |Z| = \sqrt{real^2 + imaginary^2} \$

and

\$ \theta = arctan\left( \dfrac{imaginary}{real} \right) \$

Parallel impedances are found using the product-over-sum method i.e. multiply resistance by reactance and then divide by the sum of resistance and reactance: -

Therefore Z = \$\dfrac{j\omega L\times R}{j\omega L+ R}\$

But you need to take a few more steps to get a numerical answer. The above becomes: -

\$\dfrac{j\omega LR}{j\omega L+ R}\times \dfrac{R-j\omega L}{R-j\omega L}=\$

\$\dfrac{(j\omega LR)(R-j\omega L)}{R^2 + \omega^2L^2}\$ = \$\dfrac{j\omega LR^2+\omega^2 L^2R}{R^2 + \omega^2L^2}\$

At this point note that there is no j term in the denominator and given your numbers the denominator is \$4^2 + 2^2\$ = 20.

To calculate the numerator plug in the numbers to get \$(2^2\cdot 4) +j(2\cdot4^2)\$ = 16+j32.

Then square each term of the numerator and take the square root of the sum leaving the magnitude of the numerator as 35.777.

Finally divide by the denominator and the impedance magnitude is 1.7888 ohms.

Or if you want to bypass all that maths do this: -

|Impedance| = \$\dfrac{1}{\sqrt{\dfrac{1}{R^2}+\dfrac{1}{\omega^2 L^2}}}\$

There is this calculator: -